52.10.9 problem 9

Internal problem ID [8403]
Book : DIFFERENTIAL EQUATIONS with Boundary Value Problems. DENNIS G. ZILL, WARREN S. WRIGHT, MICHAEL R. CULLEN. Brooks/Cole. Boston, MA. 2013. 8th edition.
Section : CHAPTER 8 SYSTEMS OF LINEAR FIRST-ORDER DIFFERENTIAL EQUATIONS. EXERCISES 8.2. Page 346
Problem number : 9
Date solved : Wednesday, March 05, 2025 at 05:44:42 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )&=-x \left (t \right )+y \left (t \right )\\ \frac {d}{d t}y \left (t \right )&=x \left (t \right )+2 y \left (t \right )+z \left (t \right )\\ \frac {d}{d t}z \left (t \right )&=3 y \left (t \right )-z \left (t \right ) \end{align*}

✓ Maple. Time used: 0.038 (sec). Leaf size: 66

ode:=[diff(x(t),t) = -x(t)+y(t), diff(y(t),t) = x(t)+2*y(t)+z(t), diff(z(t),t) = 3*y(t)-z(t)]; 
dsolve(ode);
 

\begin{align*} x \left (t \right ) &= \frac {{\mathrm e}^{3 t} c_{2}}{4}+{\mathrm e}^{-t} c_{1} -c_3 \,{\mathrm e}^{-2 t} \\ y &= {\mathrm e}^{3 t} c_{2} +c_3 \,{\mathrm e}^{-2 t} \\ z \left (t \right ) &= \frac {3 \,{\mathrm e}^{3 t} c_{2}}{4}-{\mathrm e}^{-t} c_{1} -3 c_3 \,{\mathrm e}^{-2 t} \\ \end{align*}

✓ Mathematica. Time used: 0.005 (sec). Leaf size: 158

ode={D[x[t],t]==-x[t]+y[t],D[y[t],t]==x[t]+2*y[t]+z[t],D[z[t],t]==3*y[t]-z[t]}; 
ic={}; 
DSolve[{ode,ic},{x[t],y[t],z[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)\to \frac {1}{20} e^{-2 t} \left (c_1 \left (15 e^t+e^{5 t}+4\right )+4 c_2 \left (e^{5 t}-1\right )+c_3 \left (-5 e^t+e^{5 t}+4\right )\right ) \\ y(t)\to \frac {1}{5} e^{-2 t} \left (c_1 \left (e^{5 t}-1\right )+c_2 \left (4 e^{5 t}+1\right )+c_3 \left (e^{5 t}-1\right )\right ) \\ z(t)\to \frac {1}{20} e^{-2 t} \left (3 c_1 \left (-5 e^t+e^{5 t}+4\right )+12 c_2 \left (e^{5 t}-1\right )+c_3 \left (5 e^t+3 e^{5 t}+12\right )\right ) \\ \end{align*}

✓ Sympy. Time used: 0.151 (sec). Leaf size: 65

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
z = Function("z") 
ode=[Eq(x(t) - y(t) + Derivative(x(t), t),0),Eq(-x(t) - 2*y(t) - z(t) + Derivative(y(t), t),0),Eq(-3*y(t) + z(t) + Derivative(z(t), t),0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t),z(t)],ics=ics)
 

\[ \left [ x{\left (t \right )} = \frac {C_{1} e^{- 2 t}}{3} - C_{2} e^{- t} + \frac {C_{3} e^{3 t}}{3}, \ y{\left (t \right )} = - \frac {C_{1} e^{- 2 t}}{3} + \frac {4 C_{3} e^{3 t}}{3}, \ z{\left (t \right )} = C_{1} e^{- 2 t} + C_{2} e^{- t} + C_{3} e^{3 t}\right ] \]